1039 lines
25 KiB
Markdown
1039 lines
25 KiB
Markdown
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[Source](http://fermatslibrary.com/s/an-elementary-proof-of-wallis-product-formula-for-pi "Permalink to
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Fermat's Library | An elementary proof of Wallis’ product formula for pi annotated/explained version.
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")
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#
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Fermat's Library | An elementary proof of Wallis’ product formula for pi annotated/explained version.
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### Comments
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Get direct links to references, BibTeX extraction and comments on all arXiv papers with: ** [__ Librarian ][2]**
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John Wallis was an English mathematician who is given partial credi...
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This was actually the way that Wallis himself derived the formula. ...
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This is the proof present in most textbooks using integration and ...
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A partial sum with an odd number of factors is for instance $$ ...
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A partial sum with an even number of factors is for instance $$ ...
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Note that $s_{n+1}=frac{3}{2}frac{5}{4}...frac{2n+2-1}{2n+2-2}=s...
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Note that $$ frac{2j+1}{2(j+1)}frac{j+1}{i+j+1}+frac{2i+1}{...
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If we group all $R_{i,j}$ with the same $i+j$ we get the following ...
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Note that $$ frac{pi(n-1)}{2n}<W<frac{pi (n+1)}{2n} $$ ...
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![][9]
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AN ELEMENTARY PROOF OF WALLIS' PRODUCT FORMULA
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FOR PI
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JOHAN W
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¨
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ASTLUND
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Abstract. We give an elementary proof of the Wallis product formula for pi.
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The proof does not require any integr ation or trigon ome tric functio ns.
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1. The Wallis product formula
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In 1655, John Wallis wrote down the celebrated formula
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(1)
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2
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1
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·
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2
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3
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·
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4
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3
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·
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4
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5
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···=
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⇡
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2
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.
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Most textbook proofs of (1) rely on evaluation of some definite integral like
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Z
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⇡/2
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0
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(sin x)
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n
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dx
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by repeated partial integration. The topic is usually reserved for more advanced
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calculus courses. The purpose of this note is to show that (1) can be derived
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using only the mathematics taught in elementary school, that is, basic algebra, the
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Pythagorean theorem, and the formula ⇡ · r
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2
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for the area of a circle of radius r.
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Viggo Brun gives an account of Wallis' method in [1] (in Norwegian). Yaglom
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and Yaglom [2] give a beautiful proof of (1) which avoids integration but uses some
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quite sophisticated trigonometric identities.
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2. A number sequence
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We denote the Wallis product by
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(2) W =
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2
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1
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·
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2
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3
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·
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4
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3
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·
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4
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5
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··· .
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The partial products involving an even number of factors form an increasing se-
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quence, while those involving an odd number of factors form a decreasing sequence.
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We let s
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0
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= 0, s
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1
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= 1, and in general,
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s
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n
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=
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3
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2
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·
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5
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4
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···
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2n 1
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2n 2
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.
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The partial pro ducts of (2) with an odd number of factors can be written as
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2n
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s
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2
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n
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=
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2
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2
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· 4
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2
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···(2n)
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1 · 3
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2
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···(2n 1)
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2
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>W,
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while the partial pro ducts with an even number of factors are of the form
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2n 1
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s
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2
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n
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=
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2
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2
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· 4
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2
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···(2n 2)
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2
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1 · 3
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2
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···(2n 3)
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2
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· (2n 1)
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<W.
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It follows that
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(3)
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2n 1
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W
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<s
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2
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n
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<
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2n
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W
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.
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Date: February 21, 2005.
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1
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![][10]
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2 JOHAN W
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¨
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ASTLUND
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We denote the di↵erence s
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n+1
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s
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n
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by a
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n
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, and observe that
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a
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n
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= s
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n+1
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s
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n
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= s
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n
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✓
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2n +1
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2n
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1
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◆
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=
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s
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n
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2n
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=
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1
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2
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·
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3
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4
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···
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2n 1
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2n
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.
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We first derive the identity
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(4) a
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i
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a
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j
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=
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j +1
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i + j +1
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a
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i
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a
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j+1
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+
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i +1
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i + j +1
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a
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i+1
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a
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j
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.
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Proof. After the substitutions
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a
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i+1
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=
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2i +1
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2(i + 1)
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a
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i
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and
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a
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j+1
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=
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2j +1
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2(j + 1)
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a
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j
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,
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the right hand side of (4) becomes
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a
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i
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a
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j
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✓
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2j +1
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2(j + 1)
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·
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j +1
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i + j +1
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+
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2i +1
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2(i + 1)
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·
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i +1
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i + j +1
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◆
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= a
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i
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a
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j
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.
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⇤
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If we start from a
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2
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0
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and repeatedly apply (4), we obtain the identities
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(5) 1 = a
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2
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0
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= a
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0
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a
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1
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+ a
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1
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a
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0
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= a
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0
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a
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2
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+ a
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2
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1
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+ a
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2
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a
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0
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= ...
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···= a
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0
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a
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n
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+ a
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1
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a
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n1
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+ ···+ a
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n
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a
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0
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.
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Proof. By applying (4) to every term, the sum a
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0
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a
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n1
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+ ···+ a
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n1
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a
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0
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becomes
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✓
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a
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0
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a
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n
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+
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1
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n
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a
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1
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a
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n1
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◆
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+
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✓
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n 1
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n
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a
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1
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a
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n1
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+
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2
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n
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a
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2
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a
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n2
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◆
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+ ···+
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✓
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1
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n
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a
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n1
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a
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1
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+ a
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n
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a
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0
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◆
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.
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After collecting terms, this simplifies to a
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0
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a
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n
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+ ···+ a
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n
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a
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0
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. ⇤
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3. A geometric construction
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We divide the positive quarter of the x-y-plane into rectangles by drawing the
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straight lines x = s
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n
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and y = s
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n
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for all n. Let R
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i,j
|
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be the rectangle with lower
|
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left corner (s
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i
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,s
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j
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) and upper right corner (s
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i+1
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,s
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j+1
|
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). The area of R
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i,j
|
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is a
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i
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a
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j
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.
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Therefore the identity (5) states that the total area of the rectangles R
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i,j
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for which
|
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i + j = n is 1. We let P
|
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n
|
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be the polygonal region consisting of all rectangles R
|
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i,j
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for which i + j<n. Hence the area of P
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n
|
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is n (see Figure 1).
|
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The outer corners of P
|
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n
|
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are the points (s
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i
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,s
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j
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) for which i + j = n + 1. By the
|
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|
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Pythagorean theorem, the distance of such a point to the origin is
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q
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s
|
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2
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i
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+ s
|
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2
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j
|
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.
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By (3), this is bounded from above by
|
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|
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r
|
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2(i + j)
|
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W
|
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|
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=
|
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r
|
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2(n + 1)
|
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|
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W
|
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|
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.
|
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|
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|
Similarly, the inner corners of P
|
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n
|
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|
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are the points (s
|
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i
|
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+ s
|
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j
|
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|
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) for which i + j = n. The
|
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|
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|
distance of such a point to the origin is bounded from below by
|
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|
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r
|
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|
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2(i + j 1)
|
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W
|
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|
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=
|
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|
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r
|
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|
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2(n 1)
|
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|
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W
|
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|
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.
|
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|
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|
![][11]
|
|||
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|
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|
THE WALLIS PRODUCT FORMULA 3
|
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|
|||
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0
|
|||
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|
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1
|
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|
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3/2
|
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|
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15/8
|
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|
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35/16
|
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|
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.
|
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|
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.
|
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|
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.
|
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|
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01
|
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|
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3
|
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|
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2
|
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|
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15
|
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|
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8
|
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|
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35
|
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|
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16
|
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|
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···
|
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|
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R
|
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|
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0,0
|
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|
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R
|
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|
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0,1
|
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|
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R
|
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|
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1,0
|
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|
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R
|
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|
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1,1
|
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R
|
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|
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2,0
|
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|
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R
|
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|
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0,2
|
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|
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R
|
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3,0
|
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|
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R
|
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|
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2,1
|
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|
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R
|
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|
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1,2
|
|||
|
|
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|
R
|
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|
|
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|
0,3
|
|||
|
|
|||
|
Figure 1. The region P
|
|||
|
|
|||
|
4
|
|||
|
|
|||
|
of area 4.
|
|||
|
|
|||
|
Therefore P
|
|||
|
|
|||
|
n
|
|||
|
|
|||
|
contains a quarter circle of radius
|
|||
|
|
|||
|
p
|
|||
|
|
|||
|
2(n 1)/W , and is contained
|
|||
|
|
|||
|
in a quarter circle of radius
|
|||
|
|
|||
|
p
|
|||
|
|
|||
|
2(n + 1)/W . Since the area of a quarter circle of
|
|||
|
|
|||
|
radius r is equal to ⇡r
|
|||
|
|
|||
|
2
|
|||
|
|
|||
|
/4, we obtain the following bounds for the area of P
|
|||
|
|
|||
|
n
|
|||
|
|
|||
|
:
|
|||
|
|
|||
|
⇡(n 1)
|
|||
|
|
|||
|
2W
|
|||
|
|
|||
|
<n<
|
|||
|
|
|||
|
⇡(n + 1)
|
|||
|
|
|||
|
2W
|
|||
|
|
|||
|
.
|
|||
|
|
|||
|
Since this holds for every n, we c onclude that
|
|||
|
|
|||
|
W =
|
|||
|
|
|||
|
⇡
|
|||
|
|
|||
|
2
|
|||
|
|
|||
|
.
|
|||
|
|
|||
|
References
|
|||
|
|
|||
|
[1] Brun, Viggo., Wallis's og Brounckers formler for ⇡ (in Norwegian), Norsk matematisk
|
|||
|
|
|||
|
tidskrift, 33 (1951) 73–81.
|
|||
|
|
|||
|
[2] Yaglom, A. M. and Yaglom, I. M., An elementary derivation of the formulas of Wallis,
|
|||
|
|
|||
|
Leibnitz and Euler for the number ⇡ (in Russi an), Uspechi matematiceskich nauk. (N.
|
|||
|
|
|||
|
S.) 8, no. 5 (57) (1953), 181–187.
|
|||
|
|
|||
|
Please enable JavaScript to view the [comments powered by Disqus.][12]
|
|||
|
|
|||
|
__
|
|||
|
|
|||
|
Discussion
|
|||
|
|
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|
A partial sum with an odd number of factors is for instance $$ frac{2}{1}cdot frac{2}{3} cdot frac{4}{3} cdot frac{4}{5} cdot frac{6}{5} $$ which has 5 factors. As we can see for odd factors we always end up at the end with a term of the form $$ frac{2n}{2n-1} $$ In the case above $n=3$, $frac{2cdot3}{2cdot 3-1}=frac{6}{5}$. It's not difficult to see that for any odd number of factors $$ frac{2^2cdot4^2...(2n-2)^2}{1cdot3^2...(2n-1)}cdot frac{2n}{2n-1}= \ = frac{2^2cdot4^2...(2n-2)^2}{1cdot3^2...(2n-1)^2}cdot 2n = \ = frac{2n}{s_n^2} $$ Note that $s_{n+1}=frac{3}{2}frac{5}{4}...frac{2n+2-1}{2n+2-2}=s_nfrac{2n+1}{2n}$ Note that $$ frac{pi(n-1)}{2n}<W<frac{pi (n+1)}{2n} $$ It is now clear that as $n rightarrow infty$, $W rightarrow frac{pi}{2}$ It's worth taking a second look at the way $pi$ appears in the proof. As we know $pi$ is deeply interconnected with geometry and it's surprisingly refreshing how the author makes the bridge and by using the circle and its area brings $pi$ into the game. Note that $$ frac{2j+1}{2(j+1)}frac{j+1}{i+j+1}+frac{2i+1}{2(i+1)}frac{i+1}{i+j+1}= \ = frac{2j+1}{2(j+1)}frac{j+1}{i+j+1}frac{i+1}{i+1}+frac{2i+1}{2(i+1)}frac{i+1}{i+j+1}frac{j+1}{j+1}= \ = frac{2(j+1)(i+j+1)(i+1)}{2(j+1)(i+j+1)(i+1)}=1 $$ If we group all $R_{i,j}$ with the same $i+j$ we get the following "stripes" ![](http://i.imgur.com/Z3Y6kkC.png) The area of each "stripe" is by identity (5) equal to 1 $$ R_{0,0}=1 \ R_{1,0}+R_{0,1}=1 \ R_{2,0}+R_{0,2}+R_{1,1}=1 \ R_{3,0}+R_{0,3}+R_{2,1}+R_{1,2}=1 \ $$ And so it is easy to conclude that the area of $P_n=n$ A partial sum with an even number of factors is for instance $$ frac{2}{1}cdot frac{2}{3} cdot frac{4}{3} cdot frac{4}{5} $$ which has 4 factors. As we can see for even factors we always end up at the end with a term of the form $$ frac{2n-2}{2n-1} $$ In the case above $n=3$, $frac{2cdot3-2}{2cdot 3-1}=frac{4}{5}$. It's not difficult to see that for any even number of factors $$ frac{2^2cdot4^2...(2n-2)}{1cdot3^2...(2n-3)^2}cdot frac{2n-2}{2n-1}= \ = frac{2^2cdot4^2...(2n-2)^2}{1cdot3^2...(2n-3)^2}cdot frac{1}{2n-1} = \ = frac{2n-1}{s_n^2} $$ John Wallis was an English mathematician who is given partial credit for the development of infinitesimal calculus. He is also credited with introducing the symbol $infty$ for infinity. He similarly used $1/infty$ for an infinitesimal. ![](https://upload.wikimedia.org/wikipedia/commons/8/89/John_Wallis_by_Sir_Godfrey_Kneller%2C_Bt.jpg) Wallis made significant contributions to trigonometry, calculus, geometry, and the analysis of infinite series. In his Opera Mathematica I (1695) Wallis introduced the term "continued fraction". This is the proof present in most textbooks using integration and recursion. Consider $J_n=int^{pi/2}_0 cos^n (x) dx$. Integrating by parts with $u=cos^{n-1}(x)$ and $dv=cos(x)$ we have $$ int_0^{pi/2}cos^n (x) dx = (n-1)int_0^{pi/2} cos^{n-2} (x)sin^2 (x) dx = \ = (n-1)int_0^{pi/2} cos^{n-2} (x)(1-cos^2 (x)) dx= \ = (n-1)int_0^{pi/2} cos^{n-2} (x) dx - (n-1)int_0^{pi/2} cos^{n} (x) dx $$ Reorganizing the terms we have $$ nJ_n=(n-1)J_{n-2} $$ Now $J_1=1$ and $J_3=2/3$, $J_5=frac{2cdot4}{3cdot5}$ which means that $$ J_{2n+1}=frac{2cdot4...(2n-2)cdot(2n)}{1cdot 3 ... (2n-1)(2n+1)} $$ At the same time $J_2=frac{pi}{2cdot 2}$, $J_4=frac{3 cdot pi}{2cdot 4 cdot 2}$ and $J_6=frac{3cdot 5 cdot pi}{2cdot 4cdot 6cdot 2}$ which means that $$ J_{2n}=frac{3cdot 5 ... (2n-3) cdot (2n-1) cdot pi }{2 cdot 4...(2n-2)cdot (2n) cdot 2} $$ For $0leq x leq frac{pi}{2}$, $0leq cos (x) leq 1$ so $cos^{2n} (x) geq cos^{2n+1} (x)geq cos^{2n+2} (x)$ which implies that $J_{2n} geq J_{2n+1}geq J_{2n+2} $. Finally $$ 1 geq frac{J_{2n+1}}{J_{2n}} geq frac{J_{2n+2}}{J_{2n}} = frac{2n+1}{2n+2} $$ This was actually the way that Wallis himself derived the formula. He compared $int _{0}^{pi }sin ^{n}xdx$ for even and odd values of n, and noted that for large $n$, increasing $n$ by 1 results in a change that becomes ever smaller as $n$ increases. Since infinitesimal calculus as we know it did not yet exist then, and the mathematical analysis of the t
|
|||
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[10]: data:image/png;base64,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[11]: data:image/png;base64,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[12]: https://disqus.com/?ref_noscript
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[13]: http://fermatslibrary.com/about
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[14]: https://twitter.com/fermatslibrary
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[15]: mailto:team%40fermatslibrary.com
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